Question

对下列哪个整数 $b$ ， $b$ 进制数 $2021_b - 221_b$ 不能被 $3$ 整除？

For which of the following integers $b$ is the base- $b$ number $2021_b - 221_b$ not divisible by $3?$

$3$

$4$

$6$

$7$

$8$

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按进制定义，把这个差写成 $10$ 进制表达式： $2b^3 + 2b + 1 - 2b^2 - 2b - 1$ $= 2b^3 - 2b^2 = 2b^2(b - 1).$

要使它能被 $3$ 整除， $b$ 或 $b - 1$ 必须能被 $3$ 整除。

唯一两个条件都不满足的选项是 $8$ 。

所以正确答案是 E。

We can express this expression in base $10$ using the definition of bases: $2b^3 + 2b + 1 - 2b^2 - 2b - 1$ $= 2b^3 - 2b^2 = 2b^2(b - 1).$

For this to be divisible by $3,$ either $b$ or $b - 1$ must be divisible by $3.$

The only answer choice that satisfies neither of these conditions is $8.$

Thus, E is the correct answer.